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Problem Solving: Informed Search Algorithms

Edmondo Trentin, DIISM. Problem Solving: Informed Search Algorithms. Best-first search. Idea: use an evaluation function f(n) for each node n f(n) is an estimated "measure of desirability" of nodes Rule: expand most desirable unexpanded node Implementation :

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Problem Solving: Informed Search Algorithms

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  1. Edmondo Trentin, DIISM Problem Solving: Informed Search Algorithms

  2. Best-first search • Idea: use an evaluation functionf(n) for each node n • f(n) is an estimated "measure of desirability" of nodes • Rule:expand most desirable unexpanded node • Implementation: Order the nodes in fringe in decreasing order of desirability

  3. Romania with step costs in km

  4. Greedy best-first search • Evaluation function f(n) = h(n) (heuristic) • h(n) = estimate of cost from n to goal • e.g., hSLD(n) = straight-line distance from n to Bucharest • Greedy best-first search always expands the node that appears to be closest to goal

  5. Greedy best-first search example

  6. Greedy best-first search example

  7. Greedy best-first search example

  8. Greedy best-first search example

  9. Properties of greedy best-first search • Complete? No – can get stuck in loops, (e.g., in “Romania” we could have Iasi  Neamt  Iasi  Neamt  ...) • Time?O(bm) in the worst case, but a good heuristic can give dramatic improvement on average (bear in mind that m is the worst-case depth of the search graph) • Space?O(bm) -- keeps all nodes in memory (same considerations on worst/average as for the Time) • Optimal? No

  10. A* search • Idea: avoid expanding paths that are already expensive • Evaluation function f(n) = g(n) + h(n) • g(n)= cost of path from the root to node n • h(n) = heuristic (estimated cost from n to goal) • f(n)= estimated total cost of path through n to goal

  11. A* search example

  12. A* search example

  13. A* search example

  14. A* search example

  15. A* search example

  16. A* search example

  17. Admissible heuristics • A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n. • An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic • Example (Romania): hSLD(n) (never overestimates the actual road distance) • Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

  18. Optimality of A* (proof) • Suppose some suboptimal goal G2has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. • f(G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f(G) = g(G) since h(G) = 0 • f(G2) > f(G) from above

  19. Optimality of A* (proof) • Suppose some suboptimal goal G2has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. • f(G2) > f(G) from above • h(n) ≤ h*(n) since h is admissible • g(n) + h(n) ≤ g(n) + h*(n) • f(n) ≤ f(G) Hence f(G2) > f(n), and A* will never select G2 for expansion

  20. Consistent heuristics • A heuristic is consistent if for every node n, every successor n' of n generated by any action a, h(n) ≤ c(n,a,n') + h(n') • If h is consistent, we have f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ≥ g(n) + h(n) = f(n) • i.e., f(n) is non-decreasing along any path. • Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

  21. Optimality of A* • A* expands nodes in order of increasing f value • Gradually adds "f-contours" of nodes • Contour i has all nodes with f=fi, where fi < fi+1

  22. Properties of A* with Admissible Heuristic • Complete? Yes • Time? Depends on the heuristic. As a general rule, it is exponential in d • Space? Depends on the heuristic. As a general rule, A* Keeps all nodes in memory • Optimal? Yes

  23. Admissible heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? • h2(S) = ?

  24. Admissible heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) • h1(S) = ? 8 • h2(S) = ? 3+1+2+2+2+3+3+2 = 18

  25. Dominance • If h2(n) ≥ h1(n) for all n (being both admissible) then we say that h2dominatesh1 • As a consequence, h2is better for search • “Typical” search costs (average number of nodes expanded): • d=12 IDS = 3,644,035 nodes A*(h1) = 227 nodes A*(h2) = 73 nodes • d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

  26. Relaxed problems • A problem with fewer restrictions on the actions is called a relaxed problem • The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem • If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution • If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution

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